Lévy metric

Metric used in mathematics

In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy.

Definition

Let F , G : R → [ 0 , 1 ] {\displaystyle F,G:\mathbb {R} \to [0,1]} be two cumulative distribution functions. Define the Lévy distance between them to be

L ( F , G ) := inf { ε > 0 | F ( x − ε ) − ε ≤ G ( x ) ≤ F ( x + ε ) + ε , ∀ x ∈ R } . {\displaystyle L(F,G):=\inf\{\varepsilon >0|F(x-\varepsilon )-\varepsilon \leq G(x)\leq F(x+\varepsilon )+\varepsilon ,\;\forall x\in \mathbb {R} \}.}

Intuitively, if between the graphs of F and G one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to L(F, G).

A sequence of cumulative distribution functions { F n } n = 1 ∞ {\displaystyle \{F_{n}\}_{n=1}^{\infty }} weakly converges to another cumulative distribution function F {\displaystyle F} if and only if L ( F n , F ) → 0 {\displaystyle L(F_{n},F)\to 0} .

See also

References

  • V.M. Zolotarev (2001) [1994], "Lévy metric", Encyclopedia of Mathematics, EMS Press


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